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**Revision Notes on Differential Equations**

- The order of the differential equation is the order of thederivative of the highest order occurring in the differential equation.
- The degree of a differential equation is the degree of the highest order differential coefficient appearing in it subject to the condition that it can be expressed as a polynomial equation in derivatives.
- A solution in which the number of constants is equal to the order of the equation is called the general solution of a differential equation.
- Particular solutions are derived from the general solution by assigning different values to the constants of general solution.
- An ordinary differential equation (ODE) of order n is an equation of the formF(x, y, y',….., y
^{(n)}) = 0, where y is a function of x and y' denotes the first derivative of y with respect to x. - An ODE of order n is said to be linear if it is of the form a
_{n}(x)y^{(n)}+ a_{n-1}(x) y^{(n-1)}+ …. + a_{1}(x) y' + a_{0}(x) y = Q(x) - If both m
_{1}and m_{2}are constants, the expressions (D–m_{1}) (D–m_{2}) y and (D–m_{2}) (D–m_{1}) y are equivalent i.e. the expression is independent of the order of operational factors. - A differential equation of the form dy/ dx = f (ax+by+c) is solved by writing ax + by + c = t.
- A differential equation, M dx + N dy = 0, is homogeneous if replacement of x and y by λx and λy results in the original function multiplied by some power of λ, where the power of λ is called the degree of the original function.
- Homogeneous differential equations are solved by putting y = vx.
- Linear equation are of the form of dy/dx + Py = Q, where P and Q are functions of x alone, or constants.
- Linear equations are solved by substituting y =uv, where u and v are functions of x.
- The general method for finding the particular integral of any function is 1/ (D-α)x = e
^{αx}∫Xe^{-αx}dx

**Various methods of finding the particular integrals:**

1. When X = e^{ax }in f(D) y = X, where a is a constant

Then 1/f(D) e^{ax} = 1/f(a) e^{ax} , if f(a) ≠ 0 and

1/f(D) e^{ax} = x^{r}/f^{r}(a) e^{ax} , if f(a) = 0, where f(D) = (D-a)^{r}f(D)

2. To find P.I. when X = cos ax or sin ax

f (D) y = X

If f (– a^{2}) ≠ 0 then 1/f(D^{2}) sin ax = 1/f(-a^{2}) sin ax

If f (– a^{2}) = 0 then (D^{2} + a^{2}) is at least one factor of f (D^{2})

3. To find the P.I.when X = x^{m} where m ∈ N

f (D) y = x^{m}

y = 1/ f(D) x^{m}

4. To find the value of 1/f(D) e^{ax} V where ‘a’ is a constant and V is a function of x

1/f (D) .e^{ax} V = e^{ax}.1/f (D+a). V

5. To find 1/f (D). xV where V is a function of x

1/f (D).xV = [x- 1/f(D). f'(D)] 1/f(D) V

**Some Results on Tangents and Normals:**

1. The equation of the tangent at P(x, y) to the curve y= f(x) is Y – y = dy/dx .(X-x)

2. The equation of the normal at point P(x, y) to the curve y = f(x) is Y – y = [-1/ (dy/dx) ].(X – x )

3. The length of the tangent = CP =y √[1+(dx/dy)^{2}]

4. The length of the normal = PD = y √[1+(dy/dx)^{2}]

5. The length of the Cartesian sub tangent = CA = y dy/dx

6. The length of the Cartesian subnormal = AD = y dy/dx

7. The initial ordinate of the tangent = OB = y – x.dy/dx

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